Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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80 views

Factoriality of schubert cells in affine flag variety

Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one. For each $...
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1answer
256 views

Is there a flat proper morphism that is not finitely presented?

Is there an example of a flat proper morphism of schemes $X\rightarrow S$ whose fibers are geometrically connected, reduced and have dimension 1, but which is not itself finitely presented? What ...
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86 views

Topological space modeled by special topological structures

Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
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Fiber product of singular varieties

Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be ...
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147 views

$G_0(X) \cong G_0(X_{red})$ where X is a noetherian scheme

Let $\textbf {X}$ be a noetherian scheme, $\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$. We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$. Now I ...
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118 views

Relate Codimensions of Integral Schemes and their Generic Fibers

I have a question about an argument in the proof of Thm 8.2.5 (Dimension formula) from Liu's "Algebraic Geometry" (page 334): Firstly the problem is reduced to the affine case with $Y:=Spec \text{ }\...
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88 views

Units in the coordinate ring on a reductive group

Let $K$ be a field and $G$ a connected reductive group over $K$. Can we describe $K[G]^{*}$?
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604 views

Beauville-Laszlo for schemes

For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
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111 views

Riemann–Hurwitz Formula for Normal Projective Curves

My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 : QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold? During the proof we ...
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178 views

Hyperelliptic Curve (Liu's Book)

Let $Y:=\mathbb{P}^1_k$ and $X$ a hyperelliptic curve. We working with the definition 4.7 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288) Namely there exist finite separable map $\pi:...
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102 views

Question about Immersion

Let $X,Y$ Noetherian integral schemes and assume we have an immersion $$i:X \to Y$$ An immersion is according to https://stacks.math.columbia.edu/tag/01IO a composition of a closed immersion $c: X \...
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1answer
298 views

Affine cone example

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\...
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326 views

A curve is proper iff the space of global sections is finite-dimensional

Let $k$ be a field, $X\rightarrow \mathrm{Spec}\,k$ be a separated morphism of finite type of relative dimension$\leq 1$ (as defined here). Is it true that $f$ is proper iff $f_* \mathcal{O}_X$ is ...
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1answer
154 views

Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?

Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring. Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...
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239 views

How to deduce the following map between Zariski tangent spaces is surjective?

Let $f: X \rightarrow Y$ be a morphism of schemes: $X$, $Y$ are regular schemes, let $Z_1, Z_2$ be two closed regular subschemes of $Y$, let $x \in X \times_Y Z_2$ such that $y = f(x) \in Z_1 \cap Z_2$...
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72 views

Cohomology of sheaves on $X \cup_{Z} Y$

I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the ...
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191 views

Direct image functor commuting with infinite direct sum of sheaves

Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial. Let $f: X \rightarrow Y$ be a ...
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111 views

Direct sums of invertible sheaves commuting with global sections and the functor of points approach

I am looking at the Stacks Project's treatment of the functor of points for projective space. Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
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86 views

Factorizations of closed embeddings of smooth schemes

All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
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73 views

“Covering-flat” part in definition of morphism of sites

Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...
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220 views

Locally affine varieties and du Val singularities

Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated. I have a specific question about du Val singularities, but while ...
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1answer
400 views

Why care about Grothendieck topology? [closed]

Noah Schweber said here the following: Why would you want a notion of sheaf theory for objects more general than topological spaces? Well, the original motivation (to my understanding) was to ...
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1answer
308 views

Proving the representability of a functor that is covered by open subfunctors

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...
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1answer
272 views

Is a universally closed monomorphism a closed immersion?

The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...
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1answer
168 views

Glueing modules over $\{x\}\times \operatorname{Spec} R$

Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...
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64 views

Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set: $$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
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2answers
289 views

Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...
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1answer
466 views

Voisin examples in $p$-adic geometry

Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper ...
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178 views

Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
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120 views

Sheaf of Kähler Differentials is Invertible in Dense Open Subset

Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$ . Here I use following definitions: A surface (resp. curve) is a $2$ -dim (resp. $1$-dim) proper k scheme ...
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237 views

Tangent Space of Picard Scheme

Let $X$ be a scheme and it's Picard scheme $\underline{Pic}(X)$ exist. Denote by $\underline{Pic}^0(X)$ the connected component of the origin of the Picard scheme. My question is what the geometric ...
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121 views

Locus of trivialization of an extension of a vector bundle

Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$. We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
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Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

Let $k$ be a field and $A$ a noetherian local $k$-algebra. Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$. Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...
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256 views

Algebraic geometry “over the function field” of the base

This is vaguely similar to, but quite different from, this question. In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically ...
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359 views

True on stalks, false on affine opens [closed]

In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g. ...
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1answer
180 views

Number of distinct scheme structures on a set [closed]

Given a cardinal number $|X|$, how many isomorphism classes of schemes with the cardinality of the set of points equal to $|X|$ are there?
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1answer
344 views

Representability of Grassmannian functor by a scheme

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...
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121 views

Making a quasi-compact open into an affine open

Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...
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1answer
538 views

Strongly abnormal schemes

Call a scheme $Y$ proper positive-dimensional over $\mathrm{Spec}\,\mathbb{C}$ abnormal if there exists an irreducible scheme $X$ affine of finite type over $\mathrm{Spec}\,\mathbb{C}$ and a $\mathbb{...
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393 views

Why are algebraic schemes called algebraic?

In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me ...
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158 views

Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces

I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true. This leads me to the following question: ...
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449 views

A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
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1answer
388 views

The ring of global sections of a regular scheme

Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.
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2answers
504 views

Quasi-compactifying schemes

Let $X$ be a scheme. Does there exist an open immersion $X\rightarrow Y$ with $Y$ quasi-compact?
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196 views

Krull dimension of schemes locally of finite type over PID

Let $R$ be a commutative unital ring that is a PID. Assume that $R$ is not a DVR. Let $X$ be an integral scheme locally of finite type over $\mathrm{Spec}\,R$. Can the Krull dimension of $\mathcal{O}...
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271 views

Do codimension 1 subsets of a scheme cover it?

Let $X$ be an irreducible scheme. A point $p\in X$ is said to have codimension $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ if $\overline{\{p\}}$ has codimension $n$. Is it true that any point of positive ...
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0answers
122 views

Functorial subscheme structure on non-locally closed subsets

If we have a scheme and a locally closed subset of the underlying topological space, then there is a canonical way to put a scheme structure on it so that the inclusion map can be upgraded to a ...
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1answer
324 views

Are schemes obtained from reduced schemes by gluing infinitesimal stuff along a reduced closed subscheme?

Let $X$ be a scheme of finite type (say over the complex numbers). The set of points for which the local ring is reduced is then an open subset $U\subseteq X$. Is it true that there is a closed ...
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202 views

Does the structure morphism matter in GAGA?

Let $X$ be an integral Noetherian separated scheme. Can there exist two morphisms of finite type $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ such that the corresponding complex-analytic spaces are not ...
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2answers
2k views

Do Grothendieck universes matter for an algebraic geometer?

I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...

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